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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '15840.q', 'ambient_counter': 17, 'ambient_order': 15840, 'ambient_tex': 'C_2\\times M_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 58, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '15840.q.660.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '660.e1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 660, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '24.12', 'subgroup_hash': 12, 'subgroup_order': 24, 'subgroup_tex': 'S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '15840.q', 'aut_centralizer_order': None, 'aut_label': '660.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7920.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['44.a1.a1', '132.c1.a1', '330.b1.a1'], 'contains': ['1320.e1.a1', '1980.i1.a1', '2640.c1.a1'], 'core': '15840.a1.a1', 'coset_action_label': None, 'count': 330, 'diagramx': [4387, -1, 6164, -1, 3203, -1, 5007, -1], 'generators': [298419246, 251373246, 285090600, 259606086], 'label': '15840.q.660.e1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '330.b1.a1', 'old_label': '660.e1.a1', 'projective_image': '15840.q', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '660.e1.a1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[2, 4, 16, 7], [5, 3, 23, 7], [5, 4, 7, 23], [2, 8, 16, 7], [21, 19, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 1], [3, 8, 1], [4, 6, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 9, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '24.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 2], 'inner_gens': [[2, 3, 7, 16], [5, 4, 7, 23], [21, 19, 16, 7], [21, 15, 16, 7]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 2], [2, 1], [3, 2]], 'label': '24.12', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 2, 'linQ_dim': 3, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 11, 'number_subgroups': 30, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 8], [4, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 4, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2]], 'representations': {'PC': {'code': 8281755524, 'gens': [1, 2, 3, 4], 'pres': [4, -2, -3, -2, 2, 33, 146, 114, 99, 55]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [10644, 10320]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 23], 'family': 'PGL'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'SO'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'PSO'}, {'d': 3, 'q': 3, 'gens': [1557, 1699, 13205], 'family': 'PGO'}, {'d': 2, 'q': 3, 'gens': [362, 4377], 'family': 'PGU'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'CSO'}, {'d': 2, 'q': 3, 'gens': [29, 23], 'family': 'PGammaL'}, {'d': 2, 'q': 3, 'gens': [362, 4377], 'family': 'PGammaU'}, {'d': 2, 'q': 2, 'gens': [266, 337, 275], 'family': 'AGL'}, {'d': 2, 'q': 2, 'gens': [266, 337, 275], 'family': 'ASL'}, {'d': 1, 'q': 4, 'gens': [3, 7, 1], 'family': 'AGammaL'}, {'d': 2, 'q': 2, 'gens': [7, 2, 5], 'family': 'AGammaL'}, {'d': 2, 'q': 2, 'gens': [7, 2, 5], 'family': 'ASigmaL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [465, 458, 314, 124]}, 'Perm': {'d': 4, 'gens': [2, 4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 1320, 'aut_gen_orders': [11, 5, 4], 'aut_gens': [[487127904, 1046351545], [340072008, 4112779255], [261497160, 2737270831], [1522120728, 4451102689]], 'aut_group': '7920.a', 'aut_hash': 3986485404724135366, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7920, 'aut_permdeg': 165, 'aut_perms': [47907267265481101173971549131888055024170340835276204336620451611739360841579006112653262080552815204272619187295379055451158685508512858668699319091564993972770582068119488084100172746282288487961469192940998529314010267016800647142452742566187967368099082404696019935100129094498029468492873627, 48158751562725333351363769146438536282587738777483267801417619937264176889077831424375438477704235650918907367418575536960185709123835668006147095304827012640919166207187393411462519491268281239514017559803859625458531921114524159698583306919194175304195821142288771979200604681416988347185789895, 42260074587349889336386068922998332105058046044980170769717583796681836057761873331767813969192665436114016968793016683371430918116917685520291904376257469210570026627100743268344407053657177692528716523498845640016550904045831050849674263296458388165412655609328422544323403831852435139424199380], 'aut_phi_ratio': 2.0625, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 165, 1, 2], [3, 440, 1, 1], [4, 990, 1, 2], [5, 1584, 1, 1], [6, 440, 1, 1], [6, 1320, 1, 2], [8, 990, 1, 4], [10, 1584, 1, 1], [11, 720, 1, 2], [22, 720, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'M_{11}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1320, 'autcentquo_group': '7920.a', 'autcentquo_hash': 3986485404724135366, 'autcentquo_nilpotent': False, 'autcentquo_order': 7920, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'M_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 165, 2], [3, 440, 1], [4, 990, 2], [5, 1584, 1], [6, 440, 1], [6, 1320, 2], [8, 990, 4], [10, 1584, 1], [11, 720, 2], [22, 720, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '7920.a', 'commutator_count': 1, 'commutator_label': '7920.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '7920.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['7920.a', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 165, 1, 2], [3, 440, 1, 1], [4, 990, 1, 2], [5, 1584, 1, 1], [6, 440, 1, 1], [6, 1320, 1, 2], [8, 990, 2, 2], [10, 1584, 1, 1], [11, 720, 2, 1], [22, 720, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 19434, 'exponent': 1320, 'exponents_of_order': [5, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[10, 0, 2], [10, 1, 1], [11, 1, 1], [16, 0, 2], [44, 1, 1], [45, 1, 1], [55, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '15840.q', 'hash': 8034936889538388295, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1320, 'inner_gen_orders': [8, 5], 'inner_gens': [[487127904, 2582840449], [5001424686, 1046351545]], 'inner_hash': 3986485404724135366, 'inner_nilpotent': False, 'inner_order': 7920, 'inner_split': True, 'inner_tex': 'M_{11}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 10, 'irrQ_dim': 10, 'irrR_degree': 10, 'irrep_stats': [[1, 2], [10, 6], [11, 2], [16, 4], [44, 2], [45, 2], [55, 2]], 'label': '15840.q', 'linC_count': 3, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C2*M11', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 20, 'number_divisions': 16, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 114, 'number_subgroup_classes': 114, 'number_subgroups': 29116, 'old_label': None, 'order': 15840, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 331], [3, 440], [4, 1980], [5, 1584], [6, 3080], [8, 3960], [10, 1584], [11, 1440], [22, 1440]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': None, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [10, 2], [11, 2], [20, 2], [32, 2], [44, 2], [45, 2], [55, 2]], 'representations': {'Perm': {'d': 13, 'gens': [487127904, 1046351545]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times M_{11}', 'transitive_degree': 22, 'wreath_data': None, 'wreath_product': False}
